Global solvability for a class of pseudodifferential operators on the torus

TL;DR

This paper characterizes the conditions under which certain pseudodifferential operators on the torus are globally solvable, using diophantine conditions and symbol oscillation analysis.

Contribution

It provides a complete characterization of global solvability for a class of pseudodifferential operators on the torus, linking solvability to diophantine and oscillation conditions.

Findings

Characterization of solvability conditions based on diophantine properties.

Identification of super-logarithmic oscillation as a key factor.

Application to operators on the (N+1)-dimensional torus.

Abstract

We give a complete characterization for the global solvability of a pseudodifferential operator P=D_t + c(t,D_x) on the (N+1)-dimensional torus T^{N+1} = S^1_t x T^N_x. Our characterization is given in terms of diophantine conditions and a notion of super-logarithmic oscilation of the symbol of the imaginary part of c(t,D_x).